# Broker Emergence in Social Clouds

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3 selected set of brokers should satisfy the following condition: Market Accessibility: Every non-broker node should be connected to at least one broker. Our Market Accessibility condition ensures that every non-broker node can participate in a transaction either as a buyer or as a seller. Therefore our problem can be formulated as finding a Minimum Dominating Set (MDS): given a graph G(V, E) that represents a social network where vertices represent users and edges represent friendship links, finding the set of broker nodes corresponds to finding the minimum dominating set. The dominating set problem is a classical NP-complete problem and several approximation algorithms exist for finding MDS. Kuhn et al. [5] propose a distributed approximation algorithm for finding a dominating set of minimum size which means every node uses only local information when executing the algorithm. This algorithm is particularly suitable for large-scale decentralised networks and we use it here. As nodes can change their roles of buyers/sellers, a broker may be connected to buyers only or sellers only. We consider two alternatives to overcome this: (i) brokers will attempt to connect to other brokers (perturbing the original social structure); (ii) apply the (approximation) algorithm for finding a connected minimum dominating set instead of a minimum dominating set (which may not be connected). Such algorithm although not distributed is presented in Guha et al [6]. It ensures that every broker node is connected to at least one other broker node. IV. METHODOLOGY We consider a network with an associated set of peer-nodes P={p 1, p 2, p 3,..., p n }, and a sub-set S={p 1, p 2, p 3,..., p m }, m < n, S P, where S represents the set of non-leaf nodes from P (we exclude leaf nodes as they do not have the connectivity to act as brokers). We use two algorithms for selecting broker nodes: social score algorithm and dominating set algorithm. A. Social Score Algorithm We apply the social score selection algorithm over the set of non-leaf nodes S. The selection process can be modelled as a function f(x) : S T, where the result is a subset of peer-nodes T with the highest social score which we consider as brokers. The selection protocol for brokers is built around the notion of social score. We use social score as a metric to evaluate nodes and select brokers. Social score is calculated as an average of three metrics used to assess the connectivity of a node within a graph a node that has greater potential to link other nodes with each other has a higher social score. The metrics we use are a node s: (1) Degree Centrality, (2) Betweenness Centrality and (3) Eigenvector centrality. Within a graph G(V, E) where V is the set containing the number of vertices and E is the set Figure 1: The selection containing the number of edges, we define the following metrics: Node s degree centrality is simply the number of links incident to the node: DC(v) = deg(v) (1) Node s betweenness centrality Betweenness centrality quantifies the number of times a node acts as a bridge along the shortest path between two other nodes and is calculated as the fraction of shortest paths between node pairs that pass through the node of interest : BC(v) = p(s, t/v) ( p(s, t) ) (2) s,t V where p(s, t/v) is the number of shortest paths between users s and t that pass through node/user v, and p(s, t) is the number of all the shortest paths between the two users s and t. Node s Eigenvector centrality defines the influence of the node within a network i.e. it measures how closely a node is connected to other well connected nodes. It assigns relative scores to all nodes in the network based on the concept that connections to high-scoring nodes contribute more to the score of the node in question than equal connections to low-scoring nodes. Hence the objective is to make x i proportional to the centralities of it s n neighbours, i.e.: EC(x i ) = 1 n A i,j x j (3) λ j=1 λ is a constant. In vector notation this can be written as X = 1 λax, where λ is an eigenvalue of matrix A if there is a non-zero vector x, such that Ax = λx. Thus, we classify nodes from the perspective of their social score which is calculated as: BC + DC + EC SS = (4) 3 Figure 1 illustrates the set of non-leaf nodes for which we calculate the social scores as a basis to support broker selection. Each selected node is given a certain amount of capital which can either be the same for all nodes, or in

4 proportion to their social score. Algorithm 1 explains how brokers are selected based on the social score associated with nodes. The variable degree represents the number of current connections whereas capacity represents the maximum number of connections a node can support. This variable can be either specified in a configuration file or set as the maximum degree of the social graph: capacity = max(degree). We use a set variable for storing nodes and a marking mechanism markn ode for identifying all those brokers over a set of simulation rounds maxrounds. The algorithm starts by excluding the leaf nodes and calculating the social score for each the non-leaf nodes. The brokers are then selected as the nodes with the highest social score out of all non-leaf nodes. Algorithm 1: Brokers Selection 1: len:=node.capacity; 2: pos:=0; 3: set:=null; 4: for i := 0 to networksize by 1 do 5: len := pos; max := maxrounds; found := false; 6: while (!found) and (len>0) and (max>0) do 7: max ; r[i] := selectnewnode(); rpeer := null; 8: size:=getnodedegree(r[i]); 9: brokerobserver.calculatescore(r[i].getindex()); 10: if (r[i].isactive()) and (capacity leq r[i].capacity) then 11: rpeer := getnodeid(r[i]); 12: marknode(rpeer,r[i]); 13: addtoset(rpeer,r[i],set); 14: found:=true; 15: else 16: if (degree r[i].degree) and (r[i].degree < rpeer.gettarget()) then 17: marknode(rpeer,r[i]); 18: addtoset(rpeer,r[i],set); 19: found := true; 20: end if 21: end if 22: end while 23: if (rpeer := null) or (r[i].isbroker()) or (r[i].size rpeer.gettarget()) or (r[i].isactive()) then 24: removenode(rpeer); 25: end if 26: end for Algorithm 1 attempts to identify a minimum number of brokers within the network. The algorithm uses a classification criteria based on the social score measure introduced above: nodes with higher social score are considered better candidates as brokers. The target set of brokers is composed by the minimum set of nodes with highest social score whose total capacity is sufficient to cover all the remaining nodes (sellers and buyers). B. Dominating Set Approximation: Distributed Algorithm The dominating set distributed algorithm is an approximation algorithm for solving the dominating set problem from [5]. The algorithm relies on a linear programming (LP) formulation of the problem and consists of two parts/algorithms: first algorithm calculates the fractional solution to the LP problem and the second algorithm does the rounding part. The algorithm runs in constant time and has a provable approximation ratio. For an arbitrary possibly constant parameter k and maximum node degree, the algorithm computes the dominating set of expected size O(k 2k log( ) DS OP T ) in O(k 2 ) rounds. Where DS OP T represents the size of the optimal dominating set. Algorithm 2: LP MDS Approximation 1: x i := 0; 2: calculateδ (2) (i) ; 3: γ (2) (v i ) := δ (2) i + 1; δ(v i ) := δ i + 1; 4: for l:=k-11 to 0 by -1 do 5: ( δ(v i ) ( + 1) (l+1)/k, z i := 0 ; 6: for m:=k-1 to 0 by -1 do 7: if (δ(v i ) γ 2 (v i ) l/l+1 ) then 8: send active node to all neighbors; 9: end if 10: a(v i ) := j N i v j is activenode ; 11: if color i = gray then then 12: a(v i ) = 0; 13: end if 14: a(v i ) to all neighbors; 15: a (1) (v i ) := max j Ni a(v i ); 16: a(v i ), a (1) (v i ) ( + 1) m+1/k 17: if δ(v i ) γ (2) (v i ) l/l+1 then 18: x i := maxx i, a (1) (v i ) ( m+1 m ; 19: end if 20: send x i to all neighbors; 21: if (x j N i ) 1 then i 22: color i := gray ; 23: send color i to all neighbors; 24: δ(v i ) := j N i color j = white ; 25: end if 26: z i (1 + ( + 1) 1/k )/γ (1) (v i ) (l/l+1) 27: send δ(v i ) to all neighbors; 28: γ (1) (v i ) := max j Ni δ(v j ); 29: send γ (1) (v i ) to all neighbors; 30: γ (2) (v i ) := max j Ni γ (1) (v j ) 31: end for 32: end for The output of the algorithm is the vector x defined for all v i V which has values 0 or 1 and indicates whether a node is in the dominating set or not: if x i = 1 then node v i is in the dominating set and if x i = 0 the node is not in the dominating set. Initially each node independently runs the first part of the algorithm (Algorithm 3 from Kuhn et al. [5]) and as a result returns a fractional value between 0 and 1 for x i variables. In accordance with this approach we use the following notations. Initially all nodes are colored white. A node is colored gray if the sum of the weights of x j for v j N i exceeds 1, i.e., as soon as node is covered. The degree of a node v i is denoted δ i. The largest degree in the network graph is denoted. The notation δ (1) i = max j Ni δ j is the maximum degree of all nodes in the closed neighbourhood N i of v i. Similarly, δ (2) i = max j Ni δ (1) j is maximum degree of all nodes at distance at most two from v i. Note that it is assumed each node

6 (a) brokers Emergence (b) Degree of brokers (c) Volume of trades when increasing the number of brokers (d) Volume of trades when increasing the number of buyers and sellers (e) Volume of trades at different broker degree (f) Average revenue per broker at different broker degree (g) Number of brokers: Social Score Algorithm(h) Volume of trades: Social Score Algorithm vs. (i) Volume of trades: Social Score Algorithm vs. vs. Dominating Set Algorithm Dominating Set Algorithm Incentivising peers approach Figure 2: Summary of experimental results during simulation). After 6 simulation cycles the number of brokers within the system become stable although trading within the network still continues to take place. Experiment 2 This experiment presents how the average number of nodes connected to brokers evolves during the simulation with the initial setup provided in table I. The average number of nodes connected to brokers identify sellers and buyers within the network. From figure 2b we observe that the number of nodes connected to brokers gradually increases within the first 6 cycles of the simulation.

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