Figure 1

(Color online) An example where the search information is an important concept. (a) illustrates a visitor’s perspective of an unknown city. The visitor therefore asks a citizen with the perspective (b) of the city, or rather the higher abstraction level (c). This level is the dual map of the city, a network where streets are identified as nodes and intersections between streets as links between the nodes. We use this level to quantify the search information in (d); the average number of yes-no questions the visitor must ask the citizen to find a specific street. The necessary information to walk the shortest path from $s$ in the lower right corner to $t$ in the upper right corner is ${\log }_{2}36\text{bits}$ or roughly 6 yes/no-questions.Reuse & Permissions

Figure 2

(Color online) The information cost at each node depend on the ordering of the link. (a) The information cost does not depend on the degree if there is only one possible link. (b) It is possible to ask yes-no questions and successively eliminate groups of wrong exits if the links are ordered. Every yes-no question optimally reduces the number of possible links with $1∕2$ and the cost is ${\log }_2(k-1)$ to find the correct exit link. (c) If the links are unordered such groupings are impossible and every exit link must be considered. In such a scenario the average number of necessary yes-no questions is $(k-1)∕2$.Reuse & Permissions

Figure 3

(Color online) Search information with degenerate paths between the source $s$ and target $t$. The numbers around the nodes indicate with what probability the link is chosen on the walk to $t$. The boldfaced number is the information cost in bits at the given node. The total information cost $S(s\to t)$ above every network is given by the average cost over all paths marked with black lines (the sum of the costs at the nodes along the paths) weighted with the probability to walk the path (width of black lines). We present three scenarios. (a) The walker aims to take a specific shortest path in, the cheapest informationwise. (b) The walker chooses between two exits, both on the shortest path, randomly. This results in a lower total information cost since a random choice does not cost any information, even though some walks will go through the expensive hub to the right $\left(2.3\text{bits}\right)$. (c) The walker chooses to minimize the average information cost between $s$ and $t$. The difference between (b) and (c) is clear from the choice at node $j$. In (c) more information is used at this node to avoid the higher cost of going to the hub to the right. This is completely avoided in (a) by going to the left at $s$, but at a higher information cost.Reuse & Permissions

Figure 4

(Color online) Search information of random Erdős-Rényi (ER) networks as a function of average degree $⟨k⟩$. The number of nodes is $N={10}^3$ and we keep the networks connected. The shaded area is the contribution from degenerate paths with the upper edge corresponding to the definition of $S_u$ in Fig. 3a. The definition in Fig. 3b is inseparable in this plot from the search information according to Fig. 3c. The degenerate paths make the highly connected networks more searchable, mainly due to degenerate paths of length 2 between each pair of nodes. Thus the resulting $S$ is lower than that obtained when considering information associated to locating just one of the shortest paths (upper border of shaded area). For very low degrees, $⟨k⟩\sim 2$, the organization of the networks opens for a broad range of different topologies with very different searchability; the average shortest path increases and finally no degenerate paths exist.Reuse & Permissions

Figure 5

(Color online) $S$ as function of system size $N$ for fully connected random network topologies with fixed average degree $⟨k⟩=5$. ER refers to Erdős-Rényi random networks and SF to random scale-free networks with degree distribution $\propto 1∕{(k_0+k)}^{2.4}$ with $k_0$ adjusted for every network size so that $⟨k⟩$ is kept fixed. A star network with one node connected to the remaining nodes in the network and the remaining links randomly distributed scales asymptotically as $S={\log }_{2}N+1$ as in principle every shortest path goes through the hub with the cost ${\log }_2(N-1)$.Reuse & Permissions

Figure 6

(Color online) $S$ as a function of the exponent $\gamma$ for random scale-free networks with degree distribution $P\left(k\right)\propto 1∕k^{\gamma }$ in (a). Varying $\gamma$ implies varying average degree $⟨k⟩$ according to the second $x$-label row. An increased $\gamma$ also implies a decreased frequency of large hubs and a lengthened average shortest path between nodes. To separate the effects we in (b) study the distribution $P\left(k\right)\propto 1∕{(k_0+k)}^{\gamma }$ with $k_0$ set to keep the $⟨k⟩$ fixed, as in the insert. Here we see that even though the average shortest path increases, $S$ decreases slightly due to a decreased frequency of hubs. However, compared to (a) the average shortest path increases substantially less with increasing $\gamma$, due to the constant $⟨k⟩$. The increasing average shortest path accordingly dominates over the decreasing frequency of hubs for $\gamma >2.2$ in (a). The size of the networks is $N={10}^4$ in both (a) and (b).Reuse & Permissions

Figure 7

(Color online) $S$ versus $N$ for tree, club-tree and modular networks. Both trees have a branching ratio $d$ of 4, as seen in the illustration above. The modular network consists of communities of ten nodes, each of them connected to five other nodes. Each community is in turn connected with three other communities.Reuse & Permissions

Figure 8

(Color online) The minimum information path of length $l_{S_{\mathit{min}}}$ is the path between two nodes that regardless of distance has the smallest cost. In (a) it is clear that the right path is cheaper informationwise but longer in number of steps. (b) shows the fraction of minimum information paths that are also shortest paths in a scale-free network with $N={10}^4$ nodes and $\gamma =2.4$. Although the overall fraction is as low as 0.62, $S_{\mathit{min}}$ is only 5% smaller than $S$. As degeneracy is not considered in $S_{\mathit{min}}$ we compare with $S$ without degeneracy, $S_u$ as in Fig. 3a.Reuse & Permissions

Figure 9

(Color online) Distribution of hide $\mathcal{H}$ for nodes on various types of networks (low $\mathcal{H}$ means high visibility). We see that the Erdős-Rényi (ER) network is quite homogeneous, the scale-free (SF) network has a wider distribution, whereas the tree hierarchy is by far the least democratic in distributing the ability of different nodes to communicate. The democratic spread plotted below the box roughly estimates the division of communication in the network. The number of nodes is $N={10}^4$ for all networks.Reuse & Permissions

Figure 10

(Color online) The hierarchical features of scale-free and Erdős-Rényi networks with hide $\mathcal{H}$ representing the importance of a node. The nodes in the networks in the top of the figure are arranged in hierarchical order. The intuitively hierarchical order of the tree is reproduced with the $\mathcal{H}$ ordering and all shortest paths are hierarchical, ${\mathcal{F}}_{\mathcal{H}}=1$. The network to the right has the opposite property. A high ratio of the shortest paths are not hierarchical since a typical path repeatedly goes up and down in the $\mathcal{H}$ hierarchy. The table shows a number of real-world networks and their ${\mathcal{F}}_{\mathcal{H}}$ together with the corresponding value in randomized networks with the same degree sequence [15, 16]. The biological networks are randomized so that both bait and prey degree of all proteins are preserved. The plot in the bottom of the figure shows the behavior of scale-free and Erdős-Rényi networks with respect to ${\mathcal{F}}_{\mathcal{H}}$. Scale-free networks are $\gamma$ dependent whereas Erdős-Rényi networks are size dependent.Reuse & Permissions