A Model for Information Networks: Efficiency, Stability and Dynamics

TL;DR

This paper introduces a simple directed network model inspired by social media, analyzing its stability, efficiency, and the emergence of natural social network structures through dynamic and static game approaches.

Contribution

It develops a novel network model with non-substitutable edges, proves convergence to stable networks, and characterizes the structures that emerge under various parameters.

Findings

Asynchronous dynamics converge quickly to stable networks.

Stable networks can exhibit community and broadcast structures.

The model captures diverse social network formations.

Abstract

We introduce a simple network model that is inspired by social information networks such as twitter. Agents are nodes, connecting to another agent by building a directed edge has a cost, and reaching other agents via short directed paths has a benefit; in effect, an agent wants to reach others quickly, but without the cost of directly connecting each and every one. Even in its simplest form, edges in this framework are neither substitutes or complements in general; hence, standard techniques are required to study the model's properties and dynamics do not apply. We prove that an asynchronous edge dynamics always converge to a stable network; in fact, for this convergence is fast for a range of parameters. Moreover, the set of stable networks are nontrivial and can support the type of network structures that have been observed to appear in social information networks -- from community clusters to broadcast networks, depending on the parameters many natural formations can emerge. We further study the static game, and give classes of stable and efficient networks for nontrivial parameter ranges. We close several problems, and leave many interesting ones open.

Equations

$$U_{G,k}^s(v)=|R_{G,k}^s(v)|-c_s\cdot d{s}_G^{+}(v),$$

$$U_{G,k}^{\ell }(v)=|R_{G,k}^{\ell }(v)|-c_{\ell }\cdot d{\ell }_G^{+}(v)$$

$$U_{S_v,{\mathbf{S}}_{-v}}(v)\ge U_{S_v^{\prime },{\mathbf{S}}_{-v}}(v).$$

$$U_{\mathbf{S}-s_{uv}}(u)\le U_{\mathbf{S}}(u),U_{\mathbf{S}-{\ell }_{uv}}(u)\le U_{\mathbf{S}}(u),$$

$$U_{\mathbf{S}+s_{uv}+{\ell }_{vu}}(u)>U_{\mathbf{S}}(u)\text{mbox}then{U}_{\mathbf{S}+s_{uv}+{\ell }_{vu}}(v)<U_{\mathbf{S}}(v).$$

$$s_{vw}\in E_s\Rightarrow {\ell }_{wv}\in E_{\ell }$$

$${\ell }_{vw}\in E_{\ell }\Rightarrow s_{wv}\in E_s.$$

$$U_{G,k}^s(v)=|R_{G,k}^s(v)\cap T^s(v)|-c_s\cdot d{s}_G^{+}(v),$$

$$U_{G,k}^{\ell }(v)=|R_{G,k}^{\ell }(v)\cap T^{\ell }(v)|-c_{\ell }\cdot d{\ell }_G^{+}(v).$$

$$n(n-1)-c\left\lceil{\frac{n-1}{\big{\lfloor}\frac{k}{2}\big{\rfloor}}}\right\rceil-c(n-1)$$

$$\displaystyle(n-1)\cdot((n-1)-c)+(n-1)-c\left\lceil{\frac{n-1}{\big{\lfloor}\frac{k}{2}\big{\rfloor}}}\right\rceil$$

$$\displaystyle=n(n-1)-c\left\lceil{\frac{n-1}{\big{\lfloor}\frac{k}{2}\big{\rfloor}}}\right\rceil-c(n-1)$$

$$U_{S_v,{\mathbf{S}}_{-v}}(v)\ge U_{S_v^{\prime },{\mathbf{S}}_{-v}}(v).$$

$$\displaystyle a(a-1)-c\left\lceil{\frac{a-1}{\big{\lfloor}\nicefrac{{k}}{{2}}\big{\rfloor}}}\right\rceil-c(a-1)$$

$$\displaystyle+\;(n-a)(n-1)-c\left\lceil{\frac{n-a-1}{\big{\lfloor}\nicefrac{{k}}{{2}}\big{\rfloor}}}\right\rceil-c(n-a-1)$$

$$\displaystyle\leq n(n-1)-a(n-a)-c\left(\left\lceil{\frac{n-1}{\big{\lfloor}\nicefrac{{k}}{{2}}\big{\rfloor}}}\right\rceil-2\right)-cn$$

$$\displaystyle=n(n-1)-c\left\lceil{\frac{n-1}{\big{\lfloor}\nicefrac{{k}}{{2}}\big{\rfloor}}}\right\rceil-c(n-1)+c-a(n-a)$$

$$\displaystyle\leq n(n-1)-c\left\lceil{\frac{n-1}{\big{\lfloor}\nicefrac{{k}}{{2}}\big{\rfloor}}}\right\rceil-c(n-1)+\sqrt{n}-(n-1),$$

$$\{(x_0,x_1\dots x_D,x_1\dots x_D{x}_{D+1})|x_i\in \{0,\dots ,d\}\text{mbox}and{x}_i\ne x_{i+1}\}.$$

$$n((n-1)-c{\log }_k(n)).$$

$$a(a-1)-ca{d}_a+(n-a)(n-1)-c(n-a)d_{n-a}$$

$$=n(n-1)-a(n-a)-c\left({\log }_k(a^a{(n-a)}^{n-a})\right)$$

$$\le n(n-1)-(n-1)-c\left({\log }_k({(n-1)}^{n-1})\right)$$

$$\le n(n-1)-c{\log }_k(n^n)+c{\log }_k\left(\frac{n^n}{{(n-1)}^{n-1}}\right)-(n-1)$$

$$\le n(n-1)-c{\log }_k(n^n)$$

$$U_G^s(v)=|R_G^s(v)|-c_s\cdot d{s}_G^{+}(v)\le (n-1)-c_s,$$

$$U_G^{\ell }(v)\le (n-1)-c_{\ell }.$$

$$U_G(v)=U_G^s(v)+U_G^{\ell }(v)\le (n-1)-(c_s+c_{\ell })$$

$$\phi (G)={\sum }_v{U}_G(v)\le n\cdot ((n-1)-(c_s+c_{\ell }))$$

$$\phi (G)$$

$$={\sum }_v(n-1)-(c_s+c_{\ell }))$$

$$=n\cdot ((n-1)-(c_s+c_{\ell })).$$

$$U_{S_v,{\mathbf{S}}_{-v}}(v)\ge U_{S_v^{\prime },{\mathbf{S}}_{-v}}(v).$$

Full text

We prove that an asynchronous edge dynamics always converge to a stable network; in fact, for this convergence is fast for a range of parameters. Moreover, the set of stable networks are nontrivial and can support the type of network structures that have been observed to appear in social information networks – from community clusters to broadcast networks, depending on the parameters many natural formations can emerge. We further study the static game, and give classes of stable and efficient networks for nontrivial parameter ranges. We close several problems, and leave many interesting ones open.

1 Introduction

Online social networks such as Facebook and Twitter are now an ubiquitous part of modern life. Moreover, given the prevalence of economic situations in which the network of relationships between agents play an important role in outcomes, it is essential to rigorously understand how networks form and what network structures are likely to emerge. Large interdisciplinary subfields that combine economics, sociology, mathematics and computer science in the study of social networks are emerging (see [6] for a survey). While many models for social network exist, most are either stochastic (i.e., probabilistic models) or are learned models (i.e., constructed by fitting a set of parameters). The game theoretic approaches to network formation that exist are largely motivated by games where network infrastructure is being built and costs are shared amongst agents (see, e.g., [21] Chapter 19), and do not necessarily capture natural properties of online social networks. We introduce a simple directed network model that has a natural interpretation with respect to many online social networks. Agents are nodes in the network and the model is defined by three key parameters:

the cost $c_{s}$ of directly connecting to another agent (i.e., making a friend request), 2. 2.

the cost $c_{\ell}$ of accepting a connection another agent (i.e., confirming a friend request), and 3. 3.

the distance $k$ (i.e., maximum path length) that suffices for gaining utility from an indirect connection to another agent.

Agents trade off decisions between the cost of maintaining edges against the rewards (in terms of connectivity) from doing so. Allowing $c_{s},c_{\ell}>0$ captures many online social networks such as Facebook and LinkedIn in which one agent initiates a connection request and the other choses to accept or decline. When $c_{\ell}=0$, the model captures other online social networks such as Twitter in which a connection can be made unilaterally. The distance $k$ captures the maximum path distance that suffices for deriving utility from (indirect) connections; a generalization of this model can further allow target sets T(v) which defines the set of agents that $v$ would like to reach within distance $k$.

We study natural dynamics in which agents periodically make asynchronous decisions on whether to add or sever edges (the model and dynamics are formally introduced in Section 2). Because edges in this model are neither complements nor substitutes (see Figure 2), standard techniques for analyzing the model do not apply. However, the fact that both forces exist allows for interesting and nontrivial networks to appear as fixed points of the dynamics. In particular, as discussed in Section 3, stable networks can have both open and closed triangles (i.e., a wide range of clustering coefficients), can exhibit homophily or heterophily, and can support various classes of real-world networks (e.g., exhibiting community structures and/or features of a broadcast network).111This is in stark contrast to related settings in which the only fixedpoints are cycles and empty graphs [3]. In Section  4, we prove that the dynamic process converges; in fact when $c_{s},c_{\ell}>0$ the convergence is fast. Lastly, we prove that for a nontrivial range of parameters a flower graph is efficient and stable, and a Kautz graph is symmetric-efficient (see Section 5).

On the theoretical front, we leave open the technically challenging question of whether symmetric networks that are also stable exist for nontrivial parameter ranges. On the practical front, we leave open the question of evaluating the fit of the model against network data sets; understanding the ranges of parameters observed in real networks may lead to interesting insights about the participants and the forces that drive them.

For the sake of readability, we give an overview of the key results in the main body of the paper; the majority of the proofs are presented in the appendix.

2 Our Model & Key Definitions

2.1 Preliminaries and Notation

Let $V$ be a set of agents with $n=|V|$.

Definition 1** (Bidirected Network).**

*A bidirected network, denoted by $G=(V,E_{s},E_{\ell})$, is a network with vertex set $V$ with two types of directed edges; speaking edges $s_{uv}\in E_{s}$,222An edge $s_{uv}$ can be thought of as $u$ initiating contact with $v$. and listening edges $\ell_{vu}\in E_{\ell}$.333An edge $\ell_{vu}$ can be thought of as $v$ accepting contact from $u$. *

Note that $s_{uv}$ exists independently of $\ell_{uv}$; both, one, or none may be present in the network. When clear from context, with some abuse of notation, we drop the $s/\ell$ demarkations and simply refer to edges $uv\in E$. We let $ds^{+}_{G}(v)=|\{w|s_{vw}\in E_{s}\}|$ denote the number of outgoing speaking edges of $v$, and let $ds^{-}_{G}(v)=|\{w|s_{wv}\in E_{s}\}|$ denote the number of incoming speaking edges to $v$. The analogous definition is used for listening out-degree ($d\ell^{+}_{G}(v)$) and in-degree ($d\ell^{-}_{G}(v)$).

Definition 2** (Speaking and Listening Reachability).**

*We say there is a speaking path of length $k$ from $v$ to $u$ if there exists a set of speaking edges $s_{vv_{1}},s_{v_{1}v_{2}},\ldots,s_{v_{k-1}u}\in E_{s}$ and a set of listening edges $\ell_{uv_{k-1}},\ell_{v_{k-1}v_{k-2}},\ldots,\ell_{v_{1}v}\in E_{\ell}$. We say that a vertex $u$ that has a speaking path of length at most $k$ from $v$ is $k$-speaking-reachable from $v$, and let $R^{s}_{G,k}(v)\subseteq V$ be the set of all such vertices. Listening paths, listening reachability, and the set of listening-reachable vertices $R^{\ell}_{G,k}(v)\subseteq V$ are defined in an analogous manner. *

With some abuse of notation, when $k$ is clear from context we drop it from the notation above. Note that if $u$ is speaking-reachable from $v$ then $v$ is listening-reachable from $u$.

2.2 The Model

Each agent $v\in V$ has a strategy $S_{v}=(S^{s}_{v},S^{\ell}_{v})$, which consists of subsets of agents $S^{s}_{v},S^{\ell}_{v}\subseteq V$. Thinking of agents as vertices, $S^{s}_{v}$ (respectively $S^{\ell}_{v})$ corresponds to the set of vertices that $v$ connects to by building speaking (respectively listening) edges $s_{vu}$ (respectively $\ell_{vu}$). Thus, the strategy vector $\mathbf{S}=(S_{1},\ldots,S_{n})$ defines a bidirected network $G=(V,E_{s},E_{\ell})$ where $E_{s}=\{vu|u\in S^{s}_{v}\}$ and $E_{\ell}=\{vu|u\in S^{\ell}_{v}\}$. With some abuse of notation, we often refer to $G$ as the set of strategies and will use $G$ and $\mathbf{S}$ interchangeably.

The utility of $v$ is given by $U_{G}(v)=U^{s}_{G,k}(v)+U^{\ell}_{G,k}(v)$ where

[TABLE]

[TABLE]

are the utilities derived from speaking and listening respectively. The costs $c_{s}$ and $c_{\ell}$ capture the cost of maintaining speaking and listening edges respectively.

A natural special case is that in which one of the costs is 0 (without loss of generality $c_{\ell}=0$). For such a model, an agent can always set $S^{\ell}_{v}=V$ without loss to her utility. Hence, the strategy space boils down to $S^{s}_{v}$. Moreover, we can consider only the speaking portion of the utility $U_{G,k}(v)=U^{s}_{G,k}(v)$ without loss of generality. In such cases we drop all $s/\ell$ demarkations and simply think of a directed network $G=(V,E)=(V,E_{s})$. This special case, when we further assume that $k=\infty$, is equivalent to the network model in [3].

2.3 Dynamics

In this paper we only consider dynamics that are asynchronous (i.e., one agent updates at a time) and stochastic (i.e., agents update in a random order). A shorthand notation for the network obtained by adding (alternatively, deleting) the edge $vw$ from an existing network $G$ is $G+vw$ (alternatively, $G-vw)$. Similarly, we let $G+S_{v}$ be the network obtained by adding all $vu$ edges where $u\in S_{v}$ to $G$. The following definition is convenient for a variety of our definitions and results.

Definition 3** (Addable and Removable Edges).**

*We say an edge ${uv}\in G$ is removable if $U_{G-{uv}}(u)>U_{G}(u)$. Similarly, we say an edge $uv\not\in G$ is addable if $U_{G+{uv}}(u)>U_{G}(u)$. *

See Figure 1 for an example of a network where the addable and removable edges are depicted.

We can then define the edge dynamics as follows:

Definition 4** (Edge Dynamics).**

*In each round, one potential (speaking or listening) edge $vw$ is selected at random. Without loss of generality, assume it is a speaking edge. If $s_{vw}\in E_{s}$ then the edge is deleted if and only if it is removable. Alternatively, if $s_{vw}\notin E_{s}$ then the edge is added if and only if it is addable. The analogous definition is used for listening edges. *

2.3.1 Complementarity & Substituability

We remark that one of the difficulties in analyzing this model arises from the fact that edges can be either strategic complements or substitutes depending on the structure of the remainder of the network (see, e.g., Figure 2). Hence, standard approaches do not apply for the analysis of this model and its dynamics.

2.4 Stability and Efficiency

Denote by $\mathbf{S}_{-v}$ by the $(n-1)$-dimensional vector of the strategies played by all agents other than $v$. With some abuse of notation we use $({S_{v},\mathbf{S}_{-v}})$ and $\mathbf{S}$ and $G$ interchangeably as is convenient.

Definition 5** (Stability).**

A strategy vector $\mathbf{S}$ is said to be stable if for all agents $v$ and each potential strategy $S^{\prime}_{v}\subseteq V$ , we have that

[TABLE]

*This is equivalent to saying that $\mathbf{S}$ is a Nash equilibrium. *

In other words, no agent $v$ has any incentive to change her strategy from $S_{v}$ to $S^{\prime}_{v}$, assuming that all other agents stick to their current strategies. Observe that such a solution is self-enforcing in the sense that once the agents are playing such a solution, no one has any incentive to deviate. In fact, for our model, something stronger holds:

Proposition 6**.**

*A strategy vector is stable if and only if no edge is addable or removable. *

Pairwise stability is a common strengthening of the notion of stability. It is natural in social networks where, effectively, a link between two agents is formed only if both endpoints are in agreement, but either can unilaterally delete an edge. In our model, an agent’s utility is never decreased by an incoming edge, hence there is no difference between stability and pairwise stability. However, in the bidirected case (when $c_{s},c_{\ell}>0$), an extended notion of pairwise stability where a speaking edge and its corresponding listening edge are consider in conjunction is natural.

Definition 7** (Bidirected Pairwise Stability).**

A strategy vector $\mathbf{S}$ is said to be bi-pairwise stable if for all pairs of agents $u,v$

[TABLE]

and if

[TABLE]

For the remainder of this paper we refer to this notion as pairwise stability.

Often, notion of fairness or global optimality are important considerations. The utilitarian objective welfare of a set of strategies is the collective utility of all of the agents; i.e., for the strategy set $G$, it is $\varphi(G)=\sum_{v}U_{G}(v)$.

Definition 8** (Efficiency and Symmetry).**

*We say a set of strategies $\mathbf{S}$, and the network $G$ it defines, is efficient if it maximizes $\varphi(G)$. It is symmetric if $U_{G}(v)=U_{G}(u)$ for all $u,v\in V$, and is asymmetric otherwise. It is symmetric-efficient if it maximizes $\varphi(G)$ over to the set of symmetric networks. *

Note that, a priori, efficiency, symmetry, and stability need not be satisfied simultaneously. One aim of this work is to explore these relationships for our model.

Lastly, we give a couple of preliminary observations that will become useful in later proofs.

Lemma 9**.**

*If $G$ is bi-pairwise stable then $G$ is stable. *

Definition 10** (Complete Edges).**

With some abuse of notation, we say a speaking edge $s_{vw}$ is complete if

[TABLE]

We say a listening edge $\ell_{vw}$ is complete if

[TABLE]

In any stable or efficient network, if $c_{s},c_{\ell}>0$, all edges are complete. Hence, despite allowing unilateral actions, agreement naturally emerges.

Lemma 11**.**

*If $c_{s}>0,c_{\ell}>0$, then for any stable, pairwise stable, or efficient network $G$ all speaking and listening edges are complete. *

3 Nontrivial Network Structures

In this section we present a series of toy examples that show how a variety of nontrivial network structures are supported by our model.

3.1 Clustering

The model can support both open and closed triangles, even within the same network, for various ranges of parameters (see Figure 3). This indirectly implies that stable networks in this model may have any amount of clustering; e.g., by replicating node $z$ and with its $zu$ edge $d$ times in Figure 3, the clustering coefficient goes to 1 as $d\to 0$, and goes to 0 as $d\to\infty$.

3.2 Homophily & Heterophily

A natural generalization of our model is to define target sets $T^{s}(v)$ and $T^{\ell}(v)\subseteq V$ and consider utilities

[TABLE]

[TABLE]

Such a model captures agents who participate in a larger network, but are only interested in connecting to some subset of nodes. One could then wonder whether stable networks tend to exhibit homophily or heterophily. Interesting, even for the same set of parameters, both can be supported (see Figure 4). That being said, roughly, the smaller that $k$ is and the larger that $c_{s}$ is, the more likely homophily is to occur. E.g., in the same figure, if $k=3$ and $c_{s}>1$ then only the leftmost network is stable. In effect, the network can no longer “waste” edges that do not directly reach the target sets.

3.3 Real-World Networks Supported

We further note that a wide variety of information networks that appear in real-world datasets such as twitter can be explained by our model as the structures that they form are fixed points of the dynamics for certain ranges of parameters. Here, we briefly consider four well-known real-world networks structures (using the terminology and characterization as described in the overview by [17]) to illustrate this point.

Community Clusters

Such networks contain many different (possibly non-disjoint) communities, each of which has different interests or properties. Each node $v$ is interested in participating in some subset of these communities; these interests can be captured by $T^{s}(v)$ and $T^{\ell}(v)$ as mentioned above. The structure of a network of community clusters contains many medium or small size components that may or may not be directly connected to each other. Indeed, such structures emerge simply with the addition of the target sets as above. The size of the components varies depending on $k$.

Broadcast Networks

In these networks people are mostly connected to or through an existing central node and not so much to each other another. News media outlets and influencers on social media are examples of such central nodes. Such networks contain a main large component; reaching this component results in reaching many other nodes, and are fixed points of the dynamics for our model, when, for example, $k=\infty$, and $T(v)\approx V$ for most $v\in V$. For example, the proof of convergence for Theorem 12 shows a process that would reach some such network. The flower graph in Figure 6 is another example.

Brand Clusters

In such networks, there are many small components that are almost disconnected from each other. For an example of such networks, consider the discussions on twitter or similar social networks about local events, niche products or subjects. Individuals may mention or describe them, or connect to a small source, but do not reply to or discuss with a large group. Such structures are fixed points of our dynamics when $|T^{s}(v)|$ and $k$ are both small.

Polarized Crowds

In such networks there are a small number of main poles (e.g., republican and democrat) and the target set of each node is only one of these poles. The networks in effect look like two or more copies of a broadcast network, but split according to target set. The nodes form two large (almost) disconnected components. Such network structures are fixed points of the dynamics in our model for sufficiently large $k$, when $V$ is partitioned into (mostly) disjoint subsets such that target set of each node is the part of $V$ that it belongs to. Similar to the discussion on homophily, the smaller that $k$ is the more polarized we expect the networks to be.

4 Convergence of Dynamics

In studying the dynamics, different approaches are required for the special case where $c_{\ell}=0$ (i.e., the network is directed) and more general bidirected setting (where $c_{s},c_{\ell}>0$). In the latter, an agent’s hand is often forced; as reachability can only occur via paths of complete edges, $u$ never has incentive to add an edge $s_{uv}$ if $\ell_{vu}$ is not present (a layman’s interpretation is to say that one cannot accept a connection that is not initiated). This, in effect, speeds convergence and makes our work easier. We consider both settings in Sections 4.1 and 4.2 respectively.

4.1 The Directed Setting ($c_{\ell}=0$)

In this section we consider the setting where $c_{\ell}=0$. Recall that in this case we can assume $S_{\ell}=V$ and the model reduces to that for a directed network with only speaking edges. Moreover, for the directed setting, our results only hold for $k=\infty$. Hence, for ease of notation we drop all $s$, $\ell$ and $k$ demarkations, e.g., $G=(V,E)=(V,E_{s})$, $c=c_{s}$, $U_{G}=U_{G,k}^{s}$, and $uv=s_{uv}$.

It is a priori not clear that any dynamics should converge, or that any interesting structure could emerge at a fixed point of the dynamics.444E.g., recall that the authors of [3] find that synchronous best-response dynamics must converge to either the cycle or the empty network under the same conditions. On the other hand, as is evident in our proof of following theorem, a nontrivial class of networks are fixedpoints of the edge dynamics.

Theorem 12**.**

*The edge dynamics for $k=\infty$ converge to a stable graph. *

Proof.

In order to prove convergence, it suffices to show that for any initial network $G_{1}$, there exists a finite sequence of graphs $G_{1},\ldots,G_{s}$ such that $G_{i}\neq G_{j}$ for any $i\neq j$ and $G_{i}$ and $G_{i+1}$ differ by a single edge that is either addable or removable in $G_{i}$ such that $G_{s}$ is stable. In other words, there is a path to convergence from any state which only uses addable or removable edges. If this path is finite, there is some (albeit tiny) probability, which is bounded away from 0, that this sequence of edges is selected. Hence, it suffices to construct such a sequence. To that end let us first state some defenitions.

Definition 13**.**

*We represent each component by the set of its vertices, we call a component large if the set of its vertices is of size at least $c$ and we call it small otherwise. *

Definition 14**.**

*We call a component/vertex root if it has no incoming edge, leaf if it has no outgoing edge and isolated if it has none of them. *

We construct the sequence as follows:

Recursively delete removable edges one at a time until no removable edge remains. 2. 2.

If there is no addable edge then we have reached a stable graph. 3. 3.

Otherwise, partition the network into maximal strongly connected components; the component-level network must be a directed acyclic network (Lemma 27). It is often convenient for us to restrict to the large-components. Roots and leaves are then defined within the large-component graph. Note that there must be at least one large component, otherwise we contradict the fact that an edge is addable (Lemma 30). 4. 4.

For each root $T_{i}$ of the large-component graph, designate a special vertex $r_{i}$. Note that for any (large or small) component $C$ such that $T_{i}\not\subseteq R_{G}(C)$, the edge $xr_{i}$ is addable for any $x\in C$ (Lemma 31); hence, the edges specified in steps 5-7 must be addable. 5. 5.

If some leaf $L_{i}\neq T_{i}$ exists in the large-component network rooted at $T_{i}$, add the edge $\ell_{i}r_{i}$ for some $\ell_{i}\in L_{i}$ and go back to step 1. 6. 6.

Otherwise, all large-components have no edges between them in the large-component graph. If there are more than one large components, let $T_{1},T_{2}$ be any two large-components such that $T_{1}\not\subseteq R_{G}(T_{2})$. Add the edge $r_{2}r_{1}$. If $T_{2}\not\subseteq R_{G+r_{2}r_{1}}(T_{1})$, also add the edge $r_{1}r_{2}$. Go back to step 1. 7. 7.

Otherwise, there is exactly one large-component $T_{1}$. If there exists a small component $S_{j}$ that is a leaf, add the edge $s_{j}r_{1}$ for some $s_{j}\in S_{j}$ and go back to step 1. 8. 8.

Again, there is exactly one large-component $T_{1}$, and this set $T_{1}$ is reachable from every component (otherwise we would be in step 7). Moreover, there must be at least one small component $S_{k}$ that is a root and in which $|R_{G}(S_{k})\setminus T_{1}|>c$ (Lemma 36). Let $t_{k}r_{k}$ be an edge that reaches $T_{1}$ on a path from $S_{k}$, i.e., $r_{k}\in T_{1}$ and $t_{k}\not\in T_{1}$, and $t_{k}\in R_{G}(S_{k})$. Add the edge $r_{k}s_{k}$ for some $s_{k}\in S_{k}$ and go back to step 1.

We prove in Lemma 26 that the only addable edges are between two different components; hence only inter-component edges must be addable; the above steps cover all possible types of addable edges in sequence until none remain.

After step 5+1 (alternatively 6+1), the number of large components that existed before step 5 (alternatively 6) is reduced by at least one (see Lemma 33 and Lemma 34 respectively). Hence, we will reach step 7 in finitely many rounds.

In step 7 there is exactly one large component (see also Lemma 30), and the number components without a direct edge to $T_{1}$ is reduced by one. Moreover, after step 7 there are no removable edges, so the number of large components and the size of the largest component does not change in step 7+1 (Lemma 35). Thus, we never go back to steps 5 or 6, and after finitely many rounds, we will move on to step 8.

In step 8 again there is exactly one large component. In step 8+1 the the number of large components does not increase, thus, we never go back to step 5 or 6. While we may go back to step 7, however, every time we complete step 8 we have removed at least one edge from being addable at any point in the future (Lemma 25). Hence, we can remain in steps 7 and 8 for finitely many rounds, and the process will terminate in step 2.

One important question that remains open is the time until convergence; in particular, we conjecture that the convergence time is fast. In effect, this is equivalent to showing that there are many short paths to convergence. Proving this, however, remains a challenging technical open problem.

4.2 The Bidirected Setting $(c_{s},c_{\ell}>0)$

We now go back to the bidirected setting. As observed above, the convergence is less surprising; as edges that are not complete are easily deleted, and furthermore are never added back to the solution (see Lemma 17)

Theorem 15**.**

*If $c_{s},c_{\ell}>0$, then the edge dynamics (see Definition 4) converge to a stable network in (expected) polynomial time in the number of nodes. *

Proof.

Let $p_{G}$ be the number of complete edges (see Definition 10) and let $m_{G}$ be the number of edges in network $G$. We will prove the theorem by induction on $p_{G}+m_{G}$.

Base Case: Suppose that $p_{G}+m_{G}=0$, thus $p_{G}=0,\ m_{G}=0$ so $G$ is empty and by Proposition 37, the empty network is stable.

Inductive step: For sake of contradiction, assume that the dynamics do not converge to a stable graph. Thus, since the number of possible networks is finite, there must exist a sequence of networks that we cycle over. If there exists $v,w\in V$ such that $s_{vw}\in E_{s}$ and $\ell_{wv}\notin E_{\ell}$, then with probability at least $\dfrac{1}{2}\cdot\dfrac{1}{n}\cdot\dfrac{1}{n-1}$, the edge $s_{vw}$ is chosen. By Lemma 16, this edge is removable. Therefore, the number of edges is reduced, so $p_{G-s_{vw}}+m_{G-s_{vw}}=p_{G}+m_{G}-1$. Thus by the induction hypothesis, $G-s_{vw}$ converges to a stable network. Similarly, if $s_{vw}\notin E_{s}$ and $\ell_{wv}\in E_{\ell}$ an analogous proof follows.

Otherwise, for all ${v,w\in V}$, if $\ s_{vw}\in E_{s}$ then $\ell_{wv}\in E_{\ell}$ and if $\ell_{vw}\in E_{\ell}$ then $s_{wv}\in E_{s}$. Since $G$ is not stable, there must exist an edge which addable or removable. By Lemma 17 we know that no removed edge will be added again. Hence, the only possible change is to remove an edge. Suppose that $s_{vw}$ is an existing removable edge for $v$. The probability of selecting $s_{vw}\text{ and }\ell_{wv}$ consecutively is at least $\left(\dfrac{1}{2}\cdot\dfrac{1}{n}\cdot\dfrac{1}{n-1}\right)^{2}$. Moreover, we know that if $s_{vw}$ and $\ell_{wv}$ are selected in sequence then both will be removed as $s_{vw}$ is removable, and then $\ell_{wv}$ will be removable by Lemma 16. Thus, for new network we have that $p_{G-s_{vw}-\ell_{vw}}+m_{G-s_{vw}-\ell_{vw}}=p_{G}+m_{G}-3$.

The convergence happens in expected polynomial time because the probabilities, as computed above, shows that after $O(n^{4})$ moves in expectation, the value of $p_{G}+m_{G}$ will decrease. We know that $p_{G}+m_{G}=O(n^{2})$. Thus, after at most $O(n^{6})$ moves in expectation we will reach a stable network.

Lemma 16**.**

*Suppose that $c_{s}>0$ and $v,w\in V$. If $s_{vw}\in E_{s},\;\ell_{wv}\notin E_{\ell}$, then $s_{vw}$ is removable. *

Proof.

We know that the profit of existence a speaking edge is made by providing speaking reachability. Moreover since $\ell_{wv}\notin E_{\ell}$, by the definition of reachability, there is no path from $v$ to arbitrary node $z$ that passes through $s_{vw}$ so the gain of it zero. Therefore removing $s_{vw}$ has no loss for $v$.

Lemma 17**.**

*If $c_{s},c_{\ell}>0$ and there exist vertices $v,w\in V$ such that $s_{vw}\notin E_{s},\ \ell_{wv}\notin E_{\ell}$, then the edge dynamics will never add it back. *

Proof.

We will prove the contrapositive. Assume that some such $s_{vw}$ is added back. We know that $\ell_{wv}\notin E_{\ell}$. Thus, by the definition of reachability, there is no path from $v$ to any arbitrary node $z$ which passes through $s_{vw}$. Hence, $z\in R^{s}_{G}(v)$ if and only if $z\in R^{s}_{G+s_{vw}}(v)$. Therefore, $|R_{G}^{s}(v)|=|R_{G+s_{vw}}^{s}(v)|$. However, $ds_{G+s_{vw}}^{+}(v)=ds_{G}^{+}(v)+1$ so $U_{G+s_{vw}}(v)=U_{G}(v)-c_{s}$ and since $c_{s}>0$ thus $U_{G+s_{vw}}(v)<U_{G}(v)$. Therefore agent $v$ does not gain by adding $s_{vw}$ and wo not add it for ever. The similar proof used for listening edges.

Using this lemma, we give a very different proof than the one for the directed setting. In addition to working for arbitrary $k$, this proof also allows us to conclude that the convergence is fast.

5 Stability and Efficiency

In this section we construct classes of efficient and stable networks. We first consider nontrivial ranges of parameters $k>1$ and $c_{s},c_{\ell}\leq n-1$ (see Section 23 for a discussion of extremal parameter ranges and networks). For this range of parameters, we see interesting networks emerge in the study of efficient and stable networks.

Remark 18**.**

*For ease of notation, in the first part of this section we consider only the directed version of our model (i.e., without loss of generality, $c_{\ell}=0$). However, all results and proofs follow immediately for the bidirectional case by replacing the directed graphs in question by the analogous bidirected network where each directed edge is replaced by a complete bidirected edge. In the statement of the theorems, constraints on $c$ apply to both $c_{s}$ and $c_{\ell}$. *

The first network we consider, called balanced flower graph, is defined for $k\leq 2\sqrt{n}$ and is constructed as follows: Make a directed cycle of length \big{\lfloor}\nicefrac{{k}}{{2}}\big{\rfloor}+1 . Select one node from this set and call it the center. As long as at least \big{\lfloor}\nicefrac{{k}}{{2}}\big{\rfloor} nodes remain, select them, and, along with the center node, form another directed cycle. Repeat until fewer than \big{\lfloor}\nicefrac{{k}}{{2}}\big{\rfloor} nodes remain; then remove one non-central node from each petal (severing its edges and connecting its predecessor and successor) until you have \big{\lfloor}\nicefrac{{k}}{{2}}\big{\rfloor}-1 nodes and form them into the final petal. We denote by $q$ the number of petals. Note at most one node is removed from each petal in balancing since since $k\leq 2\sqrt{n}$, and hence q\geq\big{\lfloor}\nicefrac{{k}}{{2}}\big{\rfloor}. Note that the balanced flower network has diameter at most $k$. See Figure 6 for an example.

Theorem 19**.**

For any $4\leq k\leq 2\sqrt{n}$ and $3\leq n$, the social welfare of the balanced flower network (see Figure 6) is

[TABLE]

*Moreover, if 1\leq c<\big{\lfloor}\nicefrac{{k}}{{2}}\big{\rfloor}-1 the balanced flower network is efficient and pairwise stable. *

Before we prove the theorem, in order to show pairwise stability, a useful lemma is stated:

Lemma 20**.**

*If $G$ be stable and strongly connected, then $G$ is pairwise stable. *

Proof.

Since $G$ is stable, it contains no removable edge. As $G$ is strongly connected, for any vertex $v$, the positive component of $U_{G}(v)$ is $2(n-1)$ and cannot be increased. Hence, there is no addable pair in $G$. Thus, $G$ is pairwise stable.

of Theorem 19.

The social welfare of the balanced flower network is

[TABLE]

as the utility of the center node is (n-1)-cq=(n-1)-c\left\lceil{\frac{n-1}{\big{\lfloor}\frac{k}{2}\big{\rfloor}}}\right\rceil, and the utility of all other nodes is $(n-1)-c$. Hence, the social welfare is

[TABLE]

Let us now prove that the flower network $G$ is stable. By Lemma 20, this will give pairwise stability. We must show that for all agents $v$, and alternate strategy $s^{\prime}_{v}\subseteq V$, we have

[TABLE]

First, consider a vertex $v$ that is not the center; since all vertices are reachable from $v$ by paths of length at most $k$, the positive component of the utility cannot be increased. Hence, utility can only be improved by lowering the cost; however, $|S_{v}|=1$, and if $|S^{\prime}_{v}|=\emptyset$, then $v^{\prime}s$ utility is 0, which is strictly less than $U_{G}(v)$. Thus, $v$ does not have an alternate strategy that can improve her utility.

Now consider the center vertex $u$. Again, all vertices are reachable from $u$ by paths of length at most $k$, so the positive component of the utility cannot be increased, and the negative component can only be decreased by decreasing the number of outgoing edges from $u$. For sake of contradiction, assume there is some $|S^{\prime}_{u}|<q$ that is $u$’s best response. Note that $S^{\prime}_{u}$ must disconnect $u$ from at least one petal. Since each petal has at least \big{\lfloor}\nicefrac{{k}}{{2}}\big{\rfloor}-1 non-center nodes and c<\big{\lfloor}\nicefrac{{k}}{{2}}\big{\rfloor}-1, adding the edge from $u$ to the petal to $S^{\prime}_{u}$ would increase the utility; this contradicts the fact that $|S^{\prime}_{u}|<q$ can be a best response. Hence, the balanced flower network is stable.

Now we prove that the balanced flower network is efficient. First, consider the network that would arise if we simply connected the remainder nodes into a small petal without balancing. This is what is known as simply a flower graph. It is known that, for any $n$ and $2\leq k\leq n$, the flower network attains the fewest number of edges of any connected network on $n$ nodes with diameter $k$ (see Theorems 1 and 2 in [10]). Note that our balanced flower network has the same number of edges as the flower graph; hence it also attains the fewest number of edges of any connected network on $n$ nodes with diameter $k$. Therefore, the balanced flower network is optimal amongst the set of strongly connected graphs.

Hence, it suffices to show that a network that is not strongly connected cannot be efficient. We prove the contrapositive. Assume there is a network that is not strongly connected; we show that we can combine any two maximal strongly connected components into a single strongly connected component that is at least as efficient. Take two strongly connected of size $a\geq 1$ and $n-a\geq 1$. Clearly, from above, the social welfare of each component can only improve by making it a balanced flower, and this does not affect the social welfare of the remaining graph. Since it is possible that one component is connected to the other (but not vice versa), without loss of generality the social welfare of this network is at most

[TABLE]

where the last inequality follows as c<\big{\lfloor}\nicefrac{{k}}{{2}}\big{\rfloor}-1\leq\big{\lfloor}\sqrt{n}\big{\rfloor}-1\leq\sqrt{n} and $1\leq a\leq n-1$. For $n\geq 3$, this is less than the social welfare attained by the balanced flower.

Remark 21**.**

*In fact, for any $2\leq k$, the (unbalanced) flower network which leaves the last petal at it’s size without rebalancing is also efficient; the constraint is due to the fact that for $k>2\sqrt{n}$ balancing as described above may not be possible. We could, instead, define a recursive balancing process that continues to steal vertices (possibly several from the same petal) until a balanced flower is reached for some $k^{\prime}<k$; efficiency follows directly, as does stability for a reduced $c$ which is a function of $n$ and $k$. *

While the above network is efficient, they are highly asymmetric with a single node taking on most of the cost. Hence, we now turn our attention towards symmetric graphs, and consider a second class of graphs, known as Kautz graphs [8, 11] (see Figure 7).

The Kautz network $K_{d}^{D}$ is a directed network with $(d+1)d^{D-1}$ vertices. The vertices are labeled by strings $x_{0}\ldots x_{D-1}$ of length $D$ with $x_{i}\in\{0,\ldots,d\}$ with $x_{i}\neq x_{i+1}$. The set of edges is defined by

[TABLE]

Clearly, the network has outdegree $d$, $(d+1)d^{D}$ edges, and diameter $D+1$.

Kautz graphs arose in the study of the following question: Given a network with $n$ nodes and $m$ edges, what is the smallest possible diameter $k$? Through a series of works, it was shown that Kautz graphs are asymptotically optimal with respect to this question (see [18] for a survey).

In our case, we can rephrase the question as follows: Given a network with $n$ nodes and diameter $k$, what is the smallest possible number of edges $m$? Clearly, such a network would maximize social welfare restricted to the set of strongly connected graphs; we can extend this result to all graphs.

Theorem 22**.**

For any $k\geq 4$, $n\geq 16$, and $c<\nicefrac{{n}}{{\log_{k}(n)}}$ the Kautz network $K^{k-1}_{\log_{k}(n)}$ (see Figure 7) is asymptotically symmetric-efficient555Generally, we assume that $\log_{k}(n)\in\mathbb{Z}$, and then can make a statement about asymptotic as $n\to\infty$ for a fixed $k$ on the set of well-defined $n$. and its social welfare is

[TABLE]

*Moreover, for $c\leq 1$ the Kautz network is pairwise stable. *

Proof.

The Kautz network is strongly connected and all vertices have the same degree; hence it is symmetric. Each vertex is connected to all others and each has degree $d_{n}={\log_{k}(n)}$, hence each vertex contributes $(n-1)-c\log_{k}(n)$ to the social welfare.

As discussed in the main body of the text, the Kautz network is known to asymptotically have the fewest number of edges necessary in order to have a network of size $n$ with maximum diameter $k$ (see [18] for a survey), and hence it is asymptotically optimal amongst the set of strongly connected graphs. Analogously to the Proof of Theorem 19 (see the last paragraph), we can bootstrap this result in order to prove efficiency amongst the set of all graphs by showing that the social welfare of a network that is not strongly connected is (asymptotically) at most the social welfare of the corresponding Kautz network for this range of parameters.

We prove the contrapositive. Assume there is a network that is not strongly connected; take two strongly connected components of size $a\geq 1$ and $b=n-a$; we show that the Kautz network of size $n$ has at least as much social welfare as this graph. Clearly, from above, the social welfare of each component can only improve by making it a Kautz graph, and this does not affect the social welfare of the remaining graph. Then, (since the two components may be connected) the social welfare is at most

[TABLE]

where the last inequality follows for any constant $c$ and $k$ and large enough $n$. This is the social welfare of the Kautz graph; hence the Kautz network is asymptotically symmetric-efficient.

We now prove stability. By Lemma 20, this also gives pairwise stability. Clearly, no strategy $S_{x}^{\prime}$ such that $|S_{x}^{\prime}|\geq|S_{x}|$ can improve the utility of $x$. Hence we first consider strategies $S_{x}^{\prime}\subseteq S_{x}$. Let $x=x_{0},\ldots,x_{D}$ and $y=y_{0},\ldots,y_{D}$ be two nodes such that there is there is no $i$ such that $x_{i}=y_{D-i-1},\ldots,x_{D}=y_{0},\ldots,y_{i}$. Hence, the distance from $x$ to $y$ must be exactly $k$ in the Kautz graph. Consider any edge $xz$ such that $z_{0}\neq y_{0}$. It must again hold that there is no $i$ such that $z_{i}=y_{D-i-1},\ldots,z_{D}=y_{0},\ldots,y_{i}$; otherwise there would have been some such $i$ for $x$. Hence, the distance from $x$ to $y$ is $k+1$ if we remove edge $xx^{\prime}$ where $x^{\prime}=x_{1},\ldots,x_{D},y_{0}$. Hence, the strategy $S_{x}\setminus xx^{\prime}$ does not improve $x$’s utility since the cost reduces by at most $1$ while at least one node becomes unreachable. Note that, for every allowable $y_{0}$, some such vertex $y$ exists. Hence, no strategy $S_{x}^{\prime}\subseteq S_{x}$ improves $x$’s utility. Lastly, for sake of contradiction, assume there is an optimal strategy $S_{x}^{\prime}$ such that $|S_{x}^{\prime}|<|S_{x}|$. Then, there must be some allowable $y_{0}$ to which $x$ does not build an edge. However, as we just observed, $S_{x}^{\prime}+xy_{0}$ improves the utility of $x$. This contradicts the optimality of $S_{x}^{\prime}$. Hence, this concludes the proof.

Remark 23**.**

*We leave open the question of whether Kautz graphs are stable for any $c>1$, and more generally, the question of characterizing optimal symmetric stable graphs. *

We can further observe some properties of the extremal ranges of the parameters where the stable and efficient network structures are not as interesting. These results also lead to the following observation.

Corollary 24**.**

*The price of anarchy is 0; the price of stability is $1$. *

This follows from the definition of the price of anarchy and price of stability by combining Propositions 37 and 38 for the former and Propositions 39 and 38 for the latter.

6 Discussion & Future Work

For the directed setting $(c_{\ell}=0)$, we show that for $k=\infty$, asynchronous stochastic edge dynamics converge to a stable network; moreover these fixed points can have non-trivial network structure. Our proof does not generalize to the case of $k<\infty$, and we leave open the question of whether these dynamics converge in that setting. Proving bounds on the time to convergence and understanding the regions of attraction would also be of interest as they inform the distribution of networks we would expect to see from a generative version of this model.

For the bidirected setting, we show that for arbitrary $k$ the edge dynamics exhibit fast converge. However, developing an understanding of best-response dynamics in which, in each time step, a vertex $v$ updates her strategy $S_{v}$ in order to maximize her utility (potentially by changing multiple edges simultaneously) with respect to the current strategies $\mathbf{S}_{-v}$ of the other agents, remains open as an interesting line for future work. Our results do not directly generalize as there is no reason why a best response would be limited to only changing addable and removable edges (given two addable edges, it is not necessarily optimal to add both). For some special cases of our model, if the dynamics are performed synchronously, the results from [3] apply. Pushing this line of work further could be of interest.

With respect to the static game, we give classes of efficient networks (for $k\lesssim\sqrt{n}$) and symmetric-efficient networks (for $4\leq k$). While the former are stable for any $c_{s},c_{\ell}\lesssim\nicefrac{{k}}{{2}}$, the latter are only stable for $c_{s},c_{\ell}\leq 1$; for the latter we rely on a long line of work from combinatorics, and determining whether any symmetric stable network exists appears to be a deep and technically challenging open problem.

Lastly, an important direction for future work would be to evaluate real-world networks and see if and when this model can explain the structure. In particular, understanding the ranges of parameters observed in real networks may lead to interesting insights about the network participants and the forces that drive them.

7 Other Related Work

Due to the vast range of applications; from sociology to commerce, biology and physics, with drastically different underlying properties, many models have been developed and studied in depth (see [20] for a survey). Starting with $G(n,p)$ [9, 5], stochastic models have often taken a forefront. Depending on the observed network properties, different models take the forefront, such as preferential attachment models [4] for specific degree distributions, or small-worlds models [23] for capturing social networks. An alternate approach, is to take an existing network and fit a model using techniques from machine learning. For example, a the authors of [16] attempt to understand the Twitter network by fitting a stochastic model. However, while stochastic and learned models can explain on a macro level what is occurring in a network, on a micro level, i.e., looking at individual nodes and its edges, they remain uninformative; the motivation as to why a node would maintain an edge is abstracted away. We instead consider game theoretic models of a network in which each node is a selfish agent and decided if and whom to connect to based on her utility.

There has been a lot of very interesting work on network formation (see [13] and [21] Chapter 19 for nice surveys coming out of the Economics and Algorithmic Game Theory literature respectively). Myerson was the first to consider such models (see, e.g., [19, 2]). However, formulated the problem as a cooperative game where agents worked together towards a common goal; in our setting we assume agents have individual, or selfish, goals. In another line of work, global connection games (see [21]) are studied in which agents are not nodes in the network, rather vested parties in (individual) global connectivity properties of the game.

More closely related models consider selfish agents that are nodes in undirected networks. [15] introduced a model for the study of the (static) stability of undirected networks. Also known as the local connection game (see [21]), nodes have discounted (based on path length) rewards for being connected to another agent, and cost for making a link. Their goal was to understand the relationship between stability and efficiency, which led to further results in this direction (see, e.g., [7, 12]). The authors of [22] consider edge dynamics for this undirected model. This is further studied by [14] where it is shown that the stochastic best response dynamics may not converge; this is in contrast to our model which will always converge to a stable network.

Directed networks allow one individual to connect to another without the consent of the second individual, and thus applications are to settings such as Twitter following, while undirected network capture social networks where links are reciprocal, such as Facebook friendship.

The difference between directed and undirected graphs is not just a technicality when it comes to modeling. In undirected networks, edges are implicitly reciprocal, hence consent is required from both endpoints; thus, undirected models are suitable for many forms of economic and social relationships. For directed networks, however, a vertex can link directly to another without reciprocally; thus, directed models are more suitable for capturing interactions that are passive in one direction, as with the consumption of public content.

The closest related work to our setting, by Bala and Goyal [3], studies the stability, efficiency, and dynamics of directed networks. In fact, their can be viewed as a special case of ours when $c_{\ell}=0$ and $k=\infty$. However, their dynamics differ significantly from ours; they use lazy simultaneous best-response dynamics while we consider asynchronous stochastic edge dynamics. Due to the nature of their update, their process always converges to either a cycle or the empty network. Not only are our dynamics more natural because coordination is not required and connections are evaluated on an individual basis, but the class of networks that are fixed points of our dynamics is nontrivial.

Appendix A Convergence for directed case

Lemma 25**.**

*After step $8$, we removed at least one edge from being addable at any point in the future. Moreover, the number of large components does not increase. *

Proof.

By the same argument as in the proof of Lemma 33, we know that step $8+1$ cannot increase the number of large components.

We prove that the edge $r_{k}s_{k}$ cannot be added again in the future. Let $T_{1}$ be the largest component in $G$ (before step 8) and let $T_{1}^{\prime\prime}$ be the largest component in $G^{\prime\prime}$ (after step 8+1). We reserve $T_{1}^{\prime}$ and $G^{\prime}$ for an intermediary graphs.

It now suffices to argue that $T_{1}^{\prime\prime}\in R_{G^{\prime\prime}}(s_{k})$; hence either $s_{k}\in T_{1}^{\prime\prime}$, or $r_{k}s_{k}$ was removed in step 1 (after we added it in step 8), in which case $|R_{G^{\prime\prime}}(s_{k})\setminus T_{1}^{\prime\prime}|<c$. Since steps 7 and 8 cannot increase $R_{G^{\prime\prime}}(s_{k})$ for any future $G^{\prime\prime}$, the edge $r_{k}s_{k}$ will never become addable again.

For sake of contradiction, assume $T_{1}^{\prime\prime}\not\in R_{G^{\prime\prime}}(s_{k})$. Let $xy$ be the first removable edge such that $T_{1}^{\prime}\not\subseteq R_{G^{\prime}-xy}(s_{k})$; hence, $xy$ must have been on every path from $s_{k}$ to $T_{1}^{\prime}$, and in particular $x\in R_{G^{\prime}}(s_{k})$ and $T_{1}\not\subseteq R_{G^{\prime}-xy}(x)$. However, $|T_{1}|>c$, hence $xy$ is not removable which gives a contradiction.

Lemma 26**.**

*If $vw$ is an addable edge in $G$, then $v$ and $w$ belong to different strongly connected components of $G$. *

Proof.

An edge $vw$ is addable if $U_{G+vw}(v)>U_{G}(v)$, in other words $|R_{G+vw}(v)\setminus R_{G}(v)|\geq c$. Assume that $v$ and $w$ belong to same strongly connected component. Thus $R_{G}(v)=R_{G}(w)$, and hence $R_{G+vw}(v)=R_{G}(v)$. Therefore the positive component of the utility for $v$ does not increase, while the negative component would decrease. Hence no such $vw$ edge is addable.

Lemma 27**.**

*The component-level network is a directed acyclic graph. *

Proof.

Assume for sake of contradiction, that some set of two or more components form a cycle in the component-level graph. Then, for every $u,v$ in the cycle, there must be some path from $u$ to $v$ as there is a path between any two vertices within every component, and a path between any two components via the cycle. Hence, the cycle in fact forms a strongly connected component contradicting the maximality of the partition.

Lemma 28**.**

*If $uv$ is not removable in $G$, then $|R_{G}(v)|\geq c$. *

If $|R_{G}(v)|<c$, then by removing $uv$, the positive component of the utility decreases by less than $c$ while the negative component decreases by $c$. Hence, $uv$ is removable.

Lemma 29**.**

*If $G$ has no removable edge then any leaf component must either be an isolated vertex or large. *

Proof.

Since $G$ has no removable edge then by lemma 28 any edge $uv$ must have $|R_{G}(v)|\geq c$. Consider a leaf component $C$, and consider any edge $uv$ such that $v\in C$. If some such edge exists, then $R_{G}(v)=R_{G}(C)$, and since $C$ is a leaf $|R_{G}(C)|=|C|$. hence, $|C|>c$. If no such edge exists, then $C$ must be an isolated vertex.

Lemma 30**.**

*If $G$ has no removable edge but an addable edge, then there exists a large component in $G$. *

Proof.

There exists some edge $uv$ in $G$ that is addable. From Lemma 26, we know that $u$ and $v$ must be in different components $C_{1}$, $C_{2}$. Since $uv$ is addable $|R_{G}(C_{2})\setminus R_{G}(C_{1})|\geq c$. Toward contradiction assume that $|C_{2}|<c$. Thus, there must be at least one other component that is reachable from $C_{2}$. Thus, the component level graph has at least one leaf, and, by lemma 29, that leaf must be of size at least $c$. Thus, there exists at least one component of s ize at least $c$.

Lemma 31**.**

*For any large component $C$ and $u,v\in V$, such that $v\in C$ and $C\not\subseteq R_{G}(u)$, edge $uv$ is addable. *

Proof.

Since $C$ is large and is not reachable from $u$, then $|R_{G+uv}(u)\setminus R_{G}(u)|=|R_{G}(v)\setminus R_{G}(u)|\geq|C|\geq c$. Thus, $(u,v)$ is addable.

Lemma 32**.**

*Omitting removable edges does not increase the number of large components in graph. *

Proof.

Let $(u,v)$ be a removable edge in $G$ and let $C$ be the component that contains $u$. If $v\not\in C$, then removing $uv$ does not change the strongly connected components of $G$. Assume $v\in C$. Let $S=R_{G}(u)\setminus R_{G-uv}(u)$ be the set of vertices that are reachable from $u$ only through $uv$. Because $uv$ is removable, we know that $|S|<c$. If $C$ divided to two new components after the removal of $uv$, then $C\cap S$ and $C\setminus S$ will form two different components in $G-uv$. Moreover $|S\cap C|\leq|S|<c$. Thus we have not increased the number of large connected components in $G-uv$.

Lemma 33**.**

*After adding an edge as in step $5$ and then completing step $1$, the number of large components at the beginning of step $5$ has reduced by at least 1. *

Proof.

Clearly, adding edges cannot increase the number of components. Moreover, since $L_{i}$ is a leaf of $T_{i}$, and $L_{i}$ and $T_{i}$ there is clearly a path from every $r_{i}\in T_{i}$ to every $x_{i}\in L_{i}$ in the original graph. Using the new edge $\ell_{i}r_{i}$, there is now also a path from every $x_{i}\in L_{i}$ to every $r_{i}\in T_{i}$. Hence this reduces the number of large components by at least one. By lemma 32, omitting removable edges does not increase the number of large components in graph. Thus, overall, the number of large components reduces by at least one.

Lemma 34**.**

*After adding an edge as in step $6$ and then completing step $1$, the number of large components at the beginning of step $6$ reduces by at least one. *

Proof.

Clearly, adding edges cannot increase the number of components. Moreover, after adding $r_{2}r_{1}$ (and if necessary $r_{1}r_{2}$), the two large components $T_{1},T_{2}$ are in the same large component in the new graph. Hence, the number of large components has reduced by at least one. By lemma 32,, omitting removable edges does not increase the number of large components in graph. Thus, overall, the number of large components reduces by at least one.

Lemma 35**.**

*After adding an edge as in step $7$ and then completing step $1$, the number of large components at the beginning of step $7$ does not increase and the number of components without a direct edge to $T_{1}$ decreases by one. *

Proof.

Note that, at the beginning of step 7 there is a single large component (otherwise we would be in step 5 or 6). Clearly, adding the edge reduces the number of components without a direct edge to $T_{1}$, and adding edges cannot increase the number of components or create more roots. We now show that no edge is removable when we go to step 1, everything that holds after step 7 must also hold after step 7+1.

Let $G$ be the network at the beginning of step 7. For sake of contradiction, assume that $xy$ is a removable edge; thus, $|R_{G}+s_{j}r_{1}(x)\setminus R_{G+s_{j}r_{1}-xy}(x)|<c$. Since $s_{j}r_{1}$ was addable in $G$, we know that $xy\neq s_{j}r_{1}$. Additionally, since $s_{j}$ was an isolated vertex in $G$ we know that it has no other edge and hence $x\neq s_{j}$. Moreover, since $s_{j}$ was an isolated vertex in $G$, we know it has no incoming edge, and hence $x\not\in R_{G+sjr_{1}}(z)$ for any $z\neq x$. Thus, $R_{G+s_{j}r_{1}-xy}(x)=R_{G-xy}(x)$ and $R_{G+s_{j}r_{1}}(x)=R_{G}(x)$, and this implies that $|R_{G}(x)\setminus R_{G-xy}(x)|<c$, and hence $xy$ was removable in $G$. This gives a contradiction, as no such $xy$ exists at the beginning of step 7 (as it would have been removed in the previous step 1).

Lemma 36**.**

*There is exactly one large-component $T_{1}$, this set $T_{1}$ is reachable from every component, and there must be at least one small component $S_{k}$ with no incoming edge such that $|R_{G}(S_{k})\setminus T_{1}|>c$. *

Proof.

Exactly one large-component $T_{1}$, since at least one large component exists (Lemma 30) and if two or more exists we would be in step 5 or 6. Moreover, $T_{1}$ is reachable from every component, otherwise we would be in step 7. Lastly, we know that there must be some small component $S_{k}$ such that $|R_{G}(S_{k})\setminus T_{1}|>c$, otherwise no edge addable. In particular, any $rs_{k}$ edge with $r\in T_{1}$ and $s_{k}\in S_{k}$ is addable.

Appendix B Price of Stability and Anarchy

Proposition 37**.**

*If $c_{s}\geq 1$ or $c_{\ell}\geq 1$ then the empty network is pairwise stable. Moreover, if $c_{s}>n-1$ or $c_{\ell}>n-1$ then the only stable network (and hence the only pairwise stable graph) is the empty graph. *

of Proposition 37.

Assume without loss of generality that $c_{s}\geq 1$. We will prove the contrapositive. Assume that $G$ is not pairwise stable. Since the network is empty, there are no possible edges to remove. Thus, there must exists a pair of vertices $v,w$ such that $U^{s}_{G+s_{vw}+\ell_{wv}}(v)>U^{s}_{G}(v)$ and $U^{\ell}_{G+s_{vw}+\ell_{wv}}(w)\geq U^{\ell}_{G}(w)$. This implies that $U^{s}_{G+s_{vw}+\ell_{vw}}(v)=U^{s}_{G}(v)+1-c_{s}>U^{s}_{G}(v)$, and thus $c_{s}<1$.

Now, suppose that $c_{s}>n-1$. Note that in a stable network, the utility for any vertex $v$ is at least [math] since this can always by attained by taking the strategy $S_{v}=\emptyset$. Assume that a network $G$ is nonempty. Thus, there must exist at least one vertex $v$ such that $ds^{+}(v)>0$. Let $v$ be such a vertex. Then $U^{s}_{G}(v)=|R^{s}_{G}(v)|-c_{s}\cdot ds^{+}(v)\leq(n-1)-c_{s}<0$. Thus, $G$ is not stable.

Proposition 38**.**

*If $k=\infty$, and $0<c_{s}\leq n-1,0<c_{\ell}\leq n-1$, then the cycle is the only efficient network. *

Proof.

For ease of notation, we drop $k=\infty$ from the subscripts. For an arbitrary node $v$, if $ds^{+}_{G}(v)=0$ then there does not exist a speaking path from $v$ to rest of the network, and hence $U_{G}^{s}(v)=|R^{s}_{G}(v)|-c_{s}\cdot ds_{G}^{+}(v)=0$. Otherwise, if $ds_{G}^{+}(v)\geq 1$, then the rest of network may be reachable from $v$. Hence, $|R^{s}_{G}(v)|\leq n-1$ where equality occurs if and only if all other nodes are reachable from $v$. Therefore

[TABLE]

and we know that $c_{s}<(n-1)$. Hence, in the case of equality, the utility is improved over the setting where $ds_{G}^{+}(v)=0$. Similarly, it follows that

[TABLE]

Thus we can conclude that

[TABLE]

so

[TABLE]

By the definition of efficiency, a network is efficient if it maximizes $\varphi$ thus if we be able to find a network say $G$ such that $\varphi(G)=n\cdot((n-1)-(c_{s}+c_{\ell}))$ then we could conclude that $G$ is efficient.

Now we consider cycle $C$ to compute $\varphi(C)$. A cycle $C$ is a closed path $v_{1},v_{2},\ldots v_{n},v_{n+1}$ such that $v_{n+1}=v_{1},$ $\forall_{1\leq i\leq n}\ s_{v_{i}v_{i+1}}\in E_{s},\ \ell_{v_{i+1}v_{i}}\in E_{\ell}$. Thus by the definition of reachability for any $v,w\in C,v\in R^{s}_{G,{\infty}}(w),v\in R^{\ell}_{G,\infty}(w)$ so $|R^{s}_{G}(v)|=|R^{\ell}_{G,\infty}(v)|=n-1$. Moreover for any node $v\in C$ we have $ds_{C}^{+}(v)=d\ell_{C}^{+}(v)=1$. Therefore

[TABLE]

Now we prove that cycle is the unique efficient network. By contradiction assume that there exists another efficient network $G\neq C$ where $\varphi(G)=n\cdot((n-1)-(c_{s}+c_{\ell}))$. According to the first part of proof, the only condition for holding this equality is that for all ${v\in V}$ it holds that $ds_{G}^{+}(v)=d\ell_{G}^{+}(v)=1,\ |R^{s}_{G}(v)|=|R^{\ell}_{G}(v)|=n-1$, thus $G$ is connected in other words, for all ${v,w\in V}$, we have that $v\in R_{G}^{s}(w)$ and $v\in R_{G}^{\ell}(w)=1$. Moreover since the speaking outdegree of each node is one, so if we only consider speaking edges, we will have one cycle on the nodes. Now we want to show that listening edges are exactly in opposite direction of speaking edges. If not, there must exist pair $(v,w)$ such that $s_{vw}\in E_{s},\ell_{wv}\notin E_{\ell}$ thus by the same proof used in proposition 11, node $v$ could improve her utility by removing $s_{vw}$ and this action dos not change other nodes’ utility. This improvement in social welfare contradicts with efficiency of $G$.

Proposition 39**.**

*For $k=\infty$, if $0\leq c_{s}\leq n-1$ and $0\leq c_{\ell}\leq n-1$, then the cycle $C$ is pairwise stable. *

Proof.

We first prove stability, and then use Lemma 20 to show that pairwise stability must also hold. Using the same proof presented in Proposition 38, we know that for all ${v\in V}$, we have $|R^{s}_{G}(v)|=|R^{\ell}_{G}(v)|=n-1$ and $ds_{C}^{+}(v)=d\ell_{C}^{+}(v)=1$. To prove the stability of $C$ we must show that for all agents $v$ and any alternate strategy $S^{\prime}_{v}\subseteq V$ , we have

[TABLE]

There are only four possible moves for node $v_{i}$.

Adding $s_{v_{i}w}$ such that $w\neq v_{i+1}$ 2. 2.

Removing $s_{v_{i}v_{i+1}}$ 3. 3.

Adding $\ell_{v_{i}w}$ such that $w\neq v_{i+1}$ 4. 4.

Removing $\ell_{v_{i}v_{i+1}}$.

Now we show that after any such change, the utility of $v_{i}$ will decrease. Without loss of generality, consider a change to a speaking edge. We know that $|R^{s}_{C}(v)|=|R^{\ell}_{C}(v)|=n-1$ which is the maximum possible, so by adding a new speaking edge we could not gain; however we will loose $c_{s}$ thus $U_{C+s_{v_{i}w}}(v_{i})=U_{C}(v_{i})-c_{s}$ and since $c_{s}>0$ so $U_{C+s_{v_{i}w}}(v_{i})<U_{C}(v_{i})$. In the case of deleting, if we remove a speaking edge, then $ds_{C}^{+}(v_{i})=0$ thus for all ${w\in V,w\neq v_{i}}$ we have $w\notin R^{s}_{C}(v_{i})$ so $|R^{s}_{C}(v_{i})|=0$. Therefore, $U_{C-s_{v_{i}w}}(v_{i})=U_{C}(v_{i})+c_{s}-(n-1)$, and since $c_{s}<(n-1)$ so $U_{C-s_{v_{i}w}}(v_{i})<U_{C}(v_{i})$. The same reasoning used for listening edges. Hence the cycle is stable. By Lemma 20, it is also pairwise stable.

We conclude this section with a characterization of all efficient and stable networks for $k=1$. The proof follows directly from the definitions.

Proposition 40**.**

*Assume $k=1$. If $c_{s},c_{\ell}>1$ then the empty network is efficient and is the only stable graph. If $c_{s},c_{\ell}\leq 1$ then the complete network is efficient; if $c_{s}<1$ or $c_{s}<1$ then the complete network is the only only stable graph. If $c_{s}=c_{\ell}=1$, then all graphs are efficient and stable. *

Appendix C Other Proofs

of Lemma 9.

For sake of contradiction, assume that $G$ is pairwise stable but is not stable. Since $G$ is pairwise stable so for any speaking edge $s_{vw}\in E_{s}$ we have $U_{G}(v)\geq U_{G-s_{vw}}(v)$ and for any listening edge $\ell_{vw}\in E_{\ell}$ we have $U_{G}(v)\geq U_{G-\ell_{vw}}(v)$. Hence, $G$ has no removable edge. And because we assume that $G$ is not stable so there must exist a speaking edge $s_{vw}\not\in E_{s}$ where $U_{G}(v)<U_{G+s_{vw}}$ or a listening edge $\ell_{vw}\not\in E_{\ell}$ where $U_{G}(v)<U_{G+\ell_{vw}}(v)$ which contradicts the definition of pairwise stability.

of Lemma 11.

We will prove the contrapositive. Assume that there exist vertices $v,w\in V$ such that $s_{vw}\in E_{s}$ and $\ell_{wv}\notin E_{\ell}$. Since $\ell_{wv}\notin E_{\ell}$, if for any node $z$ we have $z\in R^{s}_{G,k}(v)$, then, by the definition of reachability, there exists a path from $v$ to $z$ that does not use edge $s_{vw}$. Thus there is no node $z$ such that $z\in R^{s}_{G,k}(v)$ and $z\not\in R^{s}_{G-s_{vw},k}(v)$. In other words $|R^{s}_{G-s_{vw},k}(v)|=|R^{s}_{G,k}(v)|$, and the positive component of the utility is the same for $v$. However, $ds^{+}_{G-s_{vw}}(v)=ds^{+}_{G}(v)-1$, so the negative component of the utility is reduced by $c_{s}$ in $G-s_{vw}$. Thus, $U_{G-s_{vw}}(v)=U_{G}(v)+c_{s}$ and, since $c_{s}>0$, we conclude that $U_{G-s_{vw}}(v)>U_{G}(v)$. Hence, $G$ is could not be stable or pairwise stable. Moreover, deleting $s_{vw}$ does not affect on the utility of other vertices so $\varphi(G-s_{vw})>\varphi(G)$. Hence $G$ is not efficient. The proof follows analogously for $\ell_{vw}$.